In this paper we investigate the response of fiber-reinforced cylindrical membranes subject to axisymmetric deformations. The membrane is considered as an incompressible material, and the phenomenon of wrinkling is taken into account by means of the relaxed energy function. Two cases are considered: transversely isotropic membranes, characterized by one family of fibers oriented in one direction, and orthotropic membranes, characterized by two family of fibers oriented in orthogonal directions. The strain-energy function is considered as the sum of two terms: The first term is associated with the isotropic properties of the base material, and the second term is used to introduce transverse isotropy or orthotropy in the mechanical response. We determine the mechanical response of the membrane as a function of fiber orientations for given boundary conditions. The objective is to find possible fiber orientations that make the membrane as stiff as possible for the given boundary conditions. Specifically, it is shown that for transversely isotropic membranes a unique fiber orientation exists, which does not affect the mechanical response, i.e., the overall behavior is identical to a nonreinforced membrane.

1.
Wainwright
,
S. A.
, 1988,
Axis and Circumference, The Cylindrical Shape of Plants and Animals
,
Harvard University Press
,
London
.
2.
Danielson
,
D. A.
, 1973, “
Human Skin as an Elastic Membrane
,”
J. Biomech.
0021-9290,
6
, pp.
539
546
.
3.
Danielson
,
D. A.
, and
Natarajan
,
S.
, 1975, “
Tension Field Theory and the Stress in Stretched Skin
,”
J. Biomech.
0021-9290,
8
, pp.
135
142
.
4.
Burton
,
K.
, and
Taylor
,
D. L.
, 1997, “
Traction Forces of Cytokinesis Measured Using Optically Modified Elastic Substrata
,”
Nature (London)
0028-0836,
385
, pp.
450
454
.
5.
Harris
,
A. K.
,
Wild
,
P.
, and
Stopak
,
D.
, 1980, “
Silicone Rubber Substrata: A New Wrinkle in Cell Locomotion
,”
Science
0036-8075,
208
, pp.
177
179
.
6.
Stafford
,
C. M.
,
Harrison
,
C.
,
Beers
,
K. L.
,
Karim
,
A.
,
Amis
,
E. J.
,
Vanlandingham
,
M. R.
,
Kim
,
H.-C.
,
Volksen
,
W.
,
Miller
,
R. D.
, and
Simonyi
,
E.
, 2004, “
A Buckling-Based Metrology for Measuring the Elastic Moduli of Polymeric Thin Films
,”
Nature Mater.
1476-1122,
3
(
8
), pp.
545
550
.
7.
Jenkins
,
C. H.
, 1996, “
Nonlinear Dynamic Response of Membranes: State of the Art—Update
,”
Appl. Mech. Rev.
0003-6900,
49
(
10
), pp.
S41
S48
.
8.
Wagner
,
H.
, 1929, “
Flat Sheet Metal Girders With Very Thin Metal Web
,”
Z. Flugtechnik Motorluftshiffahrt
,
20
, pp.
200
207
.
9.
Reissner
,
E.
, 1938, “
On Tension Field Theory
,”
Proceedings of the 5th International Congress for Applied Mechanics
,
Wiley
,
New York
, pp.
88
92
.
10.
Steigmann
,
D. J.
, 1990, “
Tension-Field Theory
,”
Proc. R. Soc. London, Ser. A
1364-5021,
429
, pp.
141
173
.
11.
Pipkin
,
A. C.
, 1986, “
The Relaxed Energy Density for Isotropic Elastic Membranes
,”
IMA J. Appl. Math.
0272-4960,
36
, pp.
85
99
.
12.
Cerda
,
E.
, and
Mahadevan
,
L.
, 2003, “
Geometry and Physics of Wrinkling
,”
Phys. Rev. Lett.
0031-9007,
90
(
7
), p.
074302
.
13.
Coman
,
C. D.
, 2007, “
Edge-Buckling in Stretched Thin Films Under In-Plane Bending
,”
ZAMP
0044-2275,
58
, pp.
510
525
.
14.
Coman
,
C. D.
, and
Haughton
,
D. M.
, 2006, “
Localized Wrinkling Instabilities in Radially Stretched Annular Thin Films
,”
Acta Mech.
0001-5970,
185
, pp.
179
200
.
15.
Wong
,
Y. W.
, and
Pellegrino
,
S.
, 2006, “
Wrinkled Membranes II: Analitical Models
,”
J. Mech. Mater. Struct.
1559-3959,
1
, pp.
25
59
.
16.
Pipkin
,
A. C.
, 1993, “
Relaxed Energy Densities for Small Deformations of Membranes
,”
IMA J. Appl. Math.
0272-4960,
50
, pp.
225
237
.
17.
Pipkin
,
A. C.
, 1994, “
Relaxed Energy Densities for Large Deformations of Membranes
,”
IMA J. Appl. Math.
0272-4960,
52
, pp.
297
308
.
18.
Epstein
,
M.
, 1999, “
On the Wrinkling of Anisotropic Elastic Membranes
,”
J. Elast.
0374-3535,
55
, pp.
99
109
.
19.
Epstein
,
M.
, and
Forcinito
,
M. A.
, 2001, “
Anisotropic Membrane Wrinkling: Theory and Analysis
,”
Int. J. Solids Struct.
0020-7683,
38
, pp.
5253
5272
.
20.
Steigmann
,
D. J.
, and
Pipkin
,
A. C.
, 1989, “
Finite Deformations of Wrinkled Membranes
,”
Q. J. Mech. Appl. Math.
0033-5614,
42
, pp.
427
440
.
21.
Ogden
,
R. W.
, 1997,
Non-Linear Elastic Deformations
,
Dover
,
New York
.
22.
Merodio
,
J.
, and
Ogden
,
R. W.
, 2005, “
Mechanical Response of Fiber-Reinforced Incompressible Non-Linearly Elastic Solids
,”
Int. J. Non-Linear Mech.
0020-7462,
40
, pp.
213
227
.
23.
Courant
,
R.
, and
Hilbert
,
D.
, 1962,
Methods of Mathematical Physics
, Vol.
2
,
Wiley
,
New York
.
You do not currently have access to this content.